R90 / G02 — audit of the combined R87 gap and R88 compiler
25 October 2026. Reviewer /root; current-agent review, not independent.
This is a new dependency audit. The two earlier investigations are not counted again.

Frozen target: campaign/originals/G02-v1.json. In particular this requires an
actual mixed quantum Gibbs state, a uniform finite-gate algorithm, all L>=3,
and exponents independent of the distance to the classical critical point.
The detailed R87 and R88 files remain the proof objects; this note records
the checks carrying the composition, rather than replacing them by a PASS.

1. The local kernel and quantum observables

The restriction includes only complete stars. Its shell physical and ancilla
labels are both conserved and must both be retained. An interior mismatch
has energy at least one, since all terms are positive and the mismatch
projector commutes with every term. On the matched sector the heat-bath
ground-state transform is exact. A shell label tau has the unique interior
zero vector sum_x sqrt(pi^tau(x))|x,x>. Unconstrained shell ancillas cause
degeneracy but do not change the conditional interior density operator.

For a doubled inner set A, tracing out its complement forces the bra and
ket complement configurations to agree. Conditioning on the immediate
physical boundary b then gives the SAME vector |phi_b> for every exterior
shell tau. Therefore rho_A^tau=sum_b pi^tau(b)|phi_b><phi_b|, including its
off-diagonal entries. The trace norm of the difference is bounded by the
classical L1 difference of these boundary marginals. Compression of an
inner observable has no off-diagonal shell-label matrix element. Thus the
claimed local indistinguishability is genuinely an all-observable bound.
It has a support prefactor; a buffer-only uniform bound is not used.

Use square balls in the infinity metric throughout the stability argument.
They are rectangles for the classical gap input; this avoids silently
importing a rectangular mixing theorem on arbitrary punctured regions.
Equivalence with graph distance changes only fixed constants. Restrictions
used below stay below L/8; small tori are treated separately.

2. The quantitative classical inputs and the rate convention

The model's Ising coupling is beta/4 and its FK parameter is
1-exp(-beta/2). For beta in a bounded interval ending at beta_c, its distance
from p_c is comparable to beta_c-beta. The classical bounds are therefore
in the correct normalization. The uniform reference gap exponent C_* can
be increased once to cover fixed and periodic boundaries and all small
volumes. No monotonicity of the Glauber gap in temperature is assumed.

The extension of the critical proof to beta<=beta_c uses only the wired
crossing UPPER bound in its disagreement exploration and block comparison.
That upper bound is monotone in the FK parameter. The positive lower
crossing bound is not used in the mixing-time upper proof. The primary
paper's lower-gap argument is a different argument and is not imported.

For the near-critical input let ell<=C d0^-2, d0=min(1,beta_c-beta).
Wired rectangle crossing decay gives a physical connection bound from an
interior point to distance u of C exp[-u/(C ell)]. Wires identifying a
boundary never count as physical paths. One can prove the point bound by
covering a square annulus by a fixed number of overlapping rectangles of
aspect ratio at most two. To handle a crossing starting near a flat side,
shift the rectangle inward; use smaller corner rectangles at a corner.
Only rectangles of fixed bounded aspect ratio are needed in the recursion.
The mixed-boundary exploration consequently gives joint boundary TV bounded
by C|A_boundary|exp[-u/(C ell)], which is the input in section 1.

The two-block gap near one, rather than just a positive constant, matters.
After each random block update, couple so that the complement of the OTHER
block agrees with probability >=1-eta. Call that flag good. From good, an
update of the other block coalesces with probability one; another update of
the same block refreshes the good flag. From bad, either update refreshes
it. The substochastic transition bound on uncoalesced bad/good states is
[[eta,1-eta],[eta/2,(1-eta)/2]]. Its nonzero eigenvalue is (1+eta)/2.
Thus rate one PER BLOCK gives continuous-time gap >=1-eta, as used in R87.
This is consistent with rate one PER SITE after the block Dirichlet-form
comparison; there is no hidden normalization by the number of sites.

Averaging cut positions in the middle third covers a site once plus
O(w/R), where w=ceil(K ell log(R+2)); eta<=R^-4. A longer-side split
shrinks the maximal side by a constant within at most two steps. Hence the
logarithm of the accumulated inverse-gap loss is bounded by
C sum_scales [ell log(R+2)/R+R^-4]. Stop at
R0=C ell[1+log(ell+2)]^2. The sum is bounded independently of the initial
volume; leaf gaps give gamma>=c d0^(4C_*). The torus reductions have only
two initial constant-overlap steps. They do not repeat a constant loss at
each scale. This checks the original uniform-volume claim, rather than a
finite-box polynomial gap whose volume dependence would ruin preparation.

3. Finite field and spectral transport: where an exponential could hide

The R83 connected commutator series is uniformly absolutely convergent for
|g| below a fixed constant. A current support of size s meets at most 5s
transverse-field stars, and each adds at most four sites. The time simplex
contributes 1/m!, leaving the rising-factorial majorant stated there. The
two Gram annihilators remain separate. The resulting perturbation has an
anchored exponential interaction bound J<=C|g| independent of volume.

For a fixed endpoint g, use H_s=Q_0+s(Q_g-Q_0) ONLY for the stability proof.
It is not the physical field path used in the compiler. Intermediate H_s
need not be frustration free. On a gap bootstrap interval its unique lowest
eigenvector has gap >=gamma/2, which suffices for the filter and spectral
transport. Its ground energy is kept as E_s and is subtracted explicitly.

The exact filters in R87/work-05 have fixed moments gamma^-k and
gamma^(-k-1). With b=a0+2d+4 and M=b+d+4, anchor counting gives
||D_s||_(F_b)<=C J gamma^(-M-1). Impose J<=c gamma^(M+1) BEFORE applying
the parameter-time Lieb-Robinson bound. Its exponential is then bounded by
a fixed constant; it is not exp(C gamma^-M). The small prefactor J survives.

The derivative of the ordinary real-time evolution has norm
C J |t|(1+|t|)^d for a fixed local observable. This follows by summing the
commutators inside its real-time light cone and the exponentially decaying
outside tail, and then integrating time. A short/long-time split of the
filter therefore gives the difference T_s-T_0 with a factor J. Bounding
T_s and T_0 separately would lose this factor and would invalidate the
stability argument. That invalid shortcut is not in the candidate.

The decomposition is, termwise,
Y_x=alpha_s T_s(h_x+sV_x)-T_0(h_x).
Every filtered operator commutes with the appropriate ground projection;
alpha_s transports that projection to P_0. Thus Y_x commutes with P_0.
Rank one then permits Z_x=Y_x-<Omega_0,Y_x Omega_0>I with Z_x P_0=0.
Summing uses T_s(H_s)=H_s, not T_s(h_x)=h_x. The stated estimate
||Z_x||+localization_tail <= C J gamma^-A, A=2M+1,
with spatial tail (1+r)^-a0 follows from the shell sums. All moment orders
and all hidden constants here are fixed independently of delta and L.

4. Local relative form bound and the small-volume crossover

Write A_r=Pi_(x,floor(r/2)) Z_x and W_r=(I-P_r)A_r(I-P_r).
The near-identity local projection step is justified by applying the
ALL-OBSERVABLE bound to A_r^*A_r, not to A_r alone. Since Z_x Omega_0=0,
its global expectation is controlled by the squared localization error.
Taking a square root gives precisely the factor
sqrt(C0)(1+r)^(d/2) exp[-r/(C xi)], alongside (1+r)^-a0.
The support prefactor has therefore been paid, not discarded.

Local projectors are nested: P_r<=P_(r-1). Consequently both W_r and
W_(r-1) annihilate P_r, and so does their difference. Its absolute quadratic
form is bounded by its norm times a^-1(1+r)^C_* times the local reference
Hamiltonian. Counting anchors introduces (1+r)^d. The polynomial part is
summable because a0>C_*+2d+10. The exponential part costs at most a fixed
power of xi; E=C_*+8 is conservative for d=2. This verifies the relative
bound b<=C J gamma^-A a^-1[1+sqrt(C0)(1+xi)^E].

At r*=floor(L/8), the residual is a sum over all anchors, so its norm has
the explicit factor L^d. It still annihilates P_0. Put
k=a0+2d+2, B=A+k+3, Y=2+gamma^-1+xi+a^-1+C0 and L0=C Y^B.
The bound exp(-u)<=C_k u^-k makes the residual <=gamma/8 when L>=L0.
Indeed its slowest algebraic factor is bounded by
C Y^(A+k+1) L^(-(a0-d)); B(a0-d)>A+k+2 pays a factor Y^-1<=gamma.
For L<L0, Weyl's inequality with ||V||<=C J L0^d applies instead.
Choose J<=c gamma L0^-d and b<=1/4. The bootstrap gap is then strictly
larger than gamma/2 in both regimes. This prevents an unproved omission
of small or intermediate tori at a delta-dependent crossover.

Putting xi<=C d0^-2 and gamma>=c d0^q0, q0=4C_*, gives fixed finite p:
p>max{q0(M+1),q0 A+2E,q0+d B max(q0,2)}.
Shrinking a fixed prefactor pays all finitely many constants. Thus the gap
claim has polynomial critical-distance dependence. It does not assert a
useful numerical exponent or a practically large field window.

5. Conditional compiler: the hypotheses now match

The preparation path first varies beta at g=0 from a Bell-product parent,
then varies g at fixed beta. The reference gap above holds uniformly on
the first segment because its critical distance is at least the final one;
the endpoint-wise stability theorem holds at every field on the second.
The compiler does NOT physically implement the nongappedness bootstrap
interpolation of section 3, and no adiabatic runtime bound is substituted.

The interaction-picture equation puts the latest integration time in the
outermost commutator. Coherent Pauli-path selection has normalization
C_A 2^72 exp(36t) q_LCU^m (13/4)_m/m!, with
q_LCU=40 t exp(10t)2^20|g|. Its moments are uniformly summable after imposing
the additional fixed bound |g|<=2^-48. The order register has O(M) entries;
support/center lists have polynomial size in M. Sorting time registers
represents the simplex average; it does not enumerate the time grid.
Non-power-of-two selection and scalar-amplitude flags must be compiled as
specified in the consolidated R88 proof. They cannot be left as free
probabilistic primitives.

Truncated Gram sums remain positive but are NOT assumed frustration free.
Their full norm error includes N. Weyl and the isolated-eigenvector bound
give a unique gapped endpoint for M=O(log(N/(epsilon gamma))). The Gaussian
flow uses energy differences, so its derivation does not assume knowledge
of the new ground energy. With the stated Fourier convention its sign
matches the negative reduced-resolvent ground-state derivative. The odd
filter has no diagonal component. Frequency and real-time truncation errors
are paid in vector norm by unitary variation of constants.

No gap for a spatially restricted parent is used. The finite-range LR rate
is O(r0^4), because both term multiplicity and term size are paid. The safe
R-4r0 separation accounts for omitted terms extending inward. Taking
ell=O(1+log(N/(epsilon gamma))), T<=C gamma^-1 ell and
R<=C gamma^-1 ell^6 controls the GLOBAL flow error, including the two N
factors inside its logarithm. The outer flow's interaction rate is O(R^4 T).
Its overlap width W<=C gamma^-6 ell^31 is polynomial, despite the growing
term support. The decomposition uses literal restrictions of these flow
terms, so the inner restricted dynamics defining each term is unchanged.

Every large patch is explicitly simulated. A selected Pauli scans only its
patch data, with a coordinate comparison against its short support list;
coherent labels do not require a free addressed memory operation. Uniform
normalizations permit coherent time parameters. Inner and outer truncated
Dyson circuits have only logarithmic dependence on their time-grid sizes.
There is no measurement-conditioned success branch. Squared internal
operator errors control clean-ancilla isometry errors without changing the
precision class. These observations justify the proposed finite-gate model.

The resource substitution is independently recomputed as
V_R<=C gamma^-2 ell^12, V_W<=C gamma^-12 ell^62,
N V_W^2 V_R T^2 <= C N gamma^-28 ell^138.
Remaining arithmetic/synthesis factors are a FIXED polynomial in b+ell.
Absorbing fixed powers of log(1/d0) into one extra power d0^-1 gives
C N max(1,delta^(-(28q0+1))) [1+log(N/epsilon)]^a poly(b).
The same generated-circuit bound pays classical operations and workspace.
Discarding the second copy and clean workspace yields the required Gibbs
mixture by trace-distance contractivity. Positive beta_min is harmless;
the preparatory beta path may include zero even though input beta does not.

6. Verdict at this checkpoint

No invalid implication was found in the detailed gap-to-compiler chain.
The clarifications about square restrictions, the two different paths,
projector nesting and all-volume crossover above are necessary when the
two candidates are combined. They do not weaken the frozen target.
The combined mathematical statement is now an original-level CANDIDATE
under current-agent review. This is not independent certification and does
not yet close R90: a second main, five auxiliaries, source comparison and
the ordinary assessment/accounting gates are still required. No original
catalog entry is changed by this checkpoint alone.
